Expansion Theorems for the G-Function. IV

MATHEMATICS

EXPANSION THEOREMS FOR THE G-FUNCTION. IV BY

C. S. MEIJER (Communicated by Prof. J. F. KoKSMA at the meeting of March 28, 1953)

Second expansion theorem First I give the following auxiliary formulae

§ 6.

I (_ I _"'_ 00

(67)

1 )"'_ 1

v

v

_,

d = F(cx) F(y-cx) v F(y)

(0

< ffi(cx) < ffi(y)),

1

(1+)

(68)

(1

v)

1

v

_,

__

dv-

2niF(y-cx)

F( 1 -cx) F(y)

(ffi(cx) < ffi(y)).

00

These formulae are well known. The first of them is equivalent to

I( 1

1- )"'-1 ,_,_1 d

u

u

u

= F(cx) F(y-cx) F(y)

(0

< ffi(cx) <

ffi(y));

0

the second may in the usual way 43 ) be derived from the first. I will now prove the following Lemma. 0

Assumptions: m, n, p and q are integers with

< n < p < q,

1

< m < q and m + n- !P- !q > 0;

the number C satisfies the inequalities C-#- 0, [arg Cl

(69)

the numbers a 1 , (3) (70)

... ,

< (m + n- !P- !q)n;

am bv ... , bm and d satisfy the conditions

ai-bh-#- 1, 2, 3, ... ffi(ai- d)< 1

(j= 1, ... ,n; h= 1, ... ,m), (j = 1, ... , n).

Assertion 1.

I(a:,-;: I 1

(71)

+>

(

Cv

av ... , aP) ( 1-v)c-d-l v-o dv bv ... , bq

00

_

2ni

- - r (1 +d-e) 43 )

Comp.

WHITTAKER

and

Qm+1.n P+ 1 .q+l

WATSON,

I

( r av ... ,aP, c) " d, bl, ... ,bq .

Modern Analysis, § 12. 22 (1927).

188

Assertion 2.

ffi(c- d)

(72)

then

If

> .0,

44 )

(73)

Proof. (74)

If m

+ n - !P - !q > 0 and at the same time

z =1=- 0 and larg zl

< (m + n- !P- !q)n,

the path of integration C in (7) may be opened out 45 ), so as to start from -ooi +a and end at ooi+ a (a is an arbitrary real number). We then obtain (75) where the contour L runs from - oo i + a to oo i + a and is so drawn that the poles ai- l, a1 - 2, ai- 3, . . . (j = l, ... , n) lie on the left of L and the poles (j = l, ... , m)

lie on the right of L. Now, whether q > p or q = p, the integral in (75) defines a function of z which is analytic 46 ) in the entire sector (74). Hence, by the theory of analytic continuation, the integral on the right hand side of (75) with q = p represents the function G~·;(z) with m + n- p > 0 in the entire sector z =1=- 0, larg zl

<

(m

+ n - p)n,

the part of this sector where Iz I > l included. 44 ) The particular case with p
189 Now the factor G;;:(Cv) of the integrand on the left of (73) is by (75) equal to (76)

Gm·" (Cv) p,q

=

__.!__, 2:rn

" m II F(1-a;+s) II F(b;-s)

~

J

q i- 1

'Pi- 1

II F(a;-s) i-m+1 II F(1-b;+s) i-n+1

L

(Cv)• ds.

Because of (70) we may choose the contour Lin such a way that ffi(s-d) < 0 for all points s on it; then we have, if the condition (72) is also satisfied, 0

<

<

ffi(c-d)

ffi(c-s).

It follows therefore from (67) that

f( 00

(77)

v

-1)c-d-1 s-c d

v

v

F(c-d) F(d-s) F(c-s) .

=

1

We now replace the function Gr;,·qn(Cv) in the integral on the left of (73) by the integral on the right hand side of (76) and change the order of the integrations 47 ); we then obtain by means of (77)

JG~:; oe

1

I

(cv al, ... ,a'P) (v-l)c-d-1 v-c dv bl, ... , bq

r

F(c-d) 2~

"'

L

n n

F(d-s)

F(c·-s)

m

F(1-a;+s) [{ F(b;-s) C' ds.

'P

II

i-n+1

F(a;--s)

q

II

i-m+1

F(1-b;+s)

The right hand side of this relation is because of (75) equal to

F(c-d) G;N-:+1

(c I~1·~ '

..

,a'P~c).

1, ... ,

q

Thus formula (73) is established. The proof of (71) is almost similar to that of (73); instead of (67) we have to use (68) 48 ). I will now prove the second expansion theorem. Assumptions: m, n, p, q and l are integers with

Theorem 4.

q > 1, 0 < n < p < q, 1 < m < q, m the numbers d1 , (78) (3)

••• ,

d1 ,

~.

• •• ,

+ n- !P- !q-! >

0 and l

> 0;

a.. and bv ... , bm satisfy the conditions

ffi(a;-dh) < 1 (j= 1, ... ,n; h= 1, ... ,Z), a;-bh::;i=1,2,3, ... (j= 1, ... ,n; h= 1, ... ,m);

the numbers A. and w satisfy the inequalities ffi(A.) >!and w ::;i: 0. This is allowed; comp. BROMWICH, Infinite Series, § 177 (1947). Because of (69) we may choose in (71) the contour (oo, 1+) in the v-plane in such a way that larg (\;v)l < (m+n-fp-iq)n for any point v on it. 47 )

48 )

I90

Assertion I. If p < q, m + n - !P- !q-! largwl < (m + n-!p-!q-!)n, then , :----am+ I !,n -

!

(79)

II F(di) p+!q+! .

(

~w

.II.

A.

l+l'f'l

r-0

0 and

Ia1, ... 'aP, Cv ..• ' c!) d d b b z,

1> ... ,

1> •• '

q

i-1 oo I --~ Ir.

>

(-r,d1,

... ,d 1;) •

Cv ... ,c!,

1/,A

(

~I-r,a1, ... ,aP)

am.n+1 w b P+1,q+1 b I 1> ... , a•

.

Assertion 2. If pO and larg wl = (m + n - !P- !q- !)n, then formula (79) is also valid, provided that the parameters d1 , . . . , d 1, a 1 , . . . , aP and bv . .. , ba fulfil not only the conditions (78) and (3) but satisfy besides the inequality (80)

ffi(d 1) >

I

q-p+

Assertion 3. values of arg w.

I ffi(!P- !q

If p

<

q

P

h-1

h-1

+I bh- I

(t

ah)

q, then formula (79) with l

=

=

I, ... , l).

0 is valid for all

Assertion 4. If q = p, m + n - p > I and larg wl < (m + n - p- !)n, then formula (79) (with q = p) is valid, provided that the parameters dv ... , d1 , av ... , a,. and bv ... , bm satisfy the conditions (78) and (3). Remark. In the cases of the assertions I, 2 and 3 the values of the manyvalued functions G'::tl::H(Aw) and G':;.'l.;i+ 1(w) are connected in the same way as the values of the functions Gr:,·;(-1w) and G':;.'l.;q\ 1(w) in the case of assertion I of theorem I. In the case of assertion 4 the values of the functions G':H·;H(Aw) and G':.t-'l.:'"i+I(w) are connected in the same way as the values of the functions G':_·~(-1w) and G':.t-'l.:'"i+ 1(w) in the case of assertion 3 of theorem I. Proof. The special case with l = 0 of theorem 4 is also a special case of theorem 2. Hence, assertion 3 and the special cases with l = 0 of the assertions I, 2 and 4 are certainly true. The cases with l = I, 2, 3, ... of the assertions I, 2 and 4 can be proved by mathematical induction. Thus we assume that arg w satisfies the inequality (8I)

Iarg w I <

(m

+n-

!P - !q -

! )n

and we suppose that the case with l = k (k > 0) of these assertions is true. Then we shall establish the truth of the case with l = k + I. For that purpose we consider first the integral

where k > 0 and r=O, I, 2, .... If we write the functionk+ 1 tfk(I/.1v) by means of (I5) as a polynomial in If.1v and after that use (68) with D~:=Ck+ 1 -dk+ 1

191 and y

= t

+ ck+V where t runs through the values 0, 1, ... , r, we find

Further we infer from (71) with C= A.w that

provided that the parameters av ... , am b1 , the conditions 49)

.•• ,

bm and d1 ,

••. ,

(3)

ai-bhoi=-1,2,3, ...

(j=1, ... ,n;h=1, ... ,m),

(85)

ai-dhoj=.1,2,3, ...

(j= 1, ... ,n;h= 1, ... ,k)

and (86)

ffi(ai- dk+l)

<

1

(j

=

dk+l fulfil

1, ... , n).

Now, let A. satisfy the inequality ffi(A.) > t. Then, if v describes the contour (=, 1 +)in the v-plane, A.v satisfies for each point v the inequality ffi(A.v) > t, provided that the radius of the circular part of the contour is sufficiently small. We therefore have by the hypothesis of the induction

{87)

provided that the parameters av ... , a'P, bv ... , bq and d1 , . . . , dk satisfy the inequalities (78) with l = k and (3) in the cases of the assertions 1 and 4 and the inequalities (78) and (80) with l = k and (3) in the case of assertion 2. We now suppose that the parameters av ... , a'P, bv ... , bq, d1 , . . . , dk 49 ) The condition (69) is certainly satisfied in the case before us, since arg w and arg ,1. fulfil the conditions (81) and (29).

192

and dk+l satisfy besides the conditions (86) and (83) in the cases of the assertions 1 and 4 and the conditions (86), (83) and (88)

1

ffi(dk+ 1 ) >

q-p+ 1

ffi(!p- !q

q

p

+ h~l ! bh- h~l ! ah)

in the case of assertion 2. Then we may because of (84) and (82) integrate either side of (87) with respect to v along the contour (oo, 1 +)and on the right hand side we may perform the integration term by term so). The result is

(89)

If (90)

ck+1 -

dk+l

=j=. 1, 2, 3, ... ,

this expansion is equivalent to (79) with l = k + l. Hence, in the cases of the assertions 1 and 4, if formula (79) with l = k is valid under the conditions (78) with l = k and (3), then formula (79) with l = k + 1 is certainly valid under the conditions (78) with l = k, (3), (86), (83) and (90). In the case of assertion 2 the validity of (79) with l = k under the conditions (78) and (80) with l = k and (3) implies its validity under the conditions (78) and {80) with l = k, (3), (86), (83), (88) and (90). Now the system of conditions (78) with l = k, (3), (86), (83) and (90) is equivalent to the system (78) with l = k + 1, (3), (83) and (90). Similarly the system (78) and (80) with l = k, (3), (86), (83), (88) and (90) is equivalent to the system (78) and (80) with l = k + 1, (3), (83) and (90). Hence, it only remains to prove that the conditions (83) and (90) may be omitted. 50) Integration term by term is certainly allowed in the cases of the assertions 1 and 4. This follows from the behaviour of the general term of the series on the right of (89) for large values of r; comp. BROMWICH, loc. cit. 47 ), § 176 B. If p < q, m+n-!-p-iq-! > 0 and Jarg wJ = (m+n-!-p-!q-i)n, the series on the right of (89) is still convergent, provided that the condition (80) with l = k + 1 is satisfied. In this case formula (89) is also valid by reasons of continuity. The case with m+n-ip-iq-,.-i = 0 and arg w = 0 may be derived from the case with m+n-!-p-iq-!- = 1 and Jarg wJ = n by means of

( Ibv .•. , bq

Qm.n z av ... , aP) p,q

=

. -1. }( e-mbm+l

-

2n~

r

·1

Qm+l,n ( ze'" av · · ·, aP) p,q

bv ... , bq

I

- e"ibm+l G';.il,n (ze-"i al, ... , aP) ~. bl, ... ,bq ~ The behaviour of the functions pcPq and G':,-; for large values of a parameter will be discussed in a paper which I hope to publish before long.

193 Now the condition (90) may be removed in the following way: We replace (82) by 51 )

valid under the conditions (83) and lR(ck+l- dk+l) Likewise we replace (84) by 52 )

>

0.

valid if the conditions (3), (85), (86) and lR(ck+l- dk+1) > 0 are satisfied. Proceeding in the same manner as before we now find that (79) with l = k + l is in the cases of the assertions l and 4 certainly valid under the conditions (78) with l = k + l, (3), (83) and lR(ck+l- dk+l) > 0; similarly in the case of assertion 2 formula (79) with l = k + l is certainly valid under the conditions (78) and (80) with l = k + l, (3), (83) and lR(ck+I- dk+l) > 0. The restriction (90) is therefore superfluous. The restriction (83) may be removed by means of the method which we used in the proof of theorem 3 in removing the restriction (54). Thus the theorem is established 53). Closing Remark. Assertion 4 of theorem 4 is equivalent to assertion 3 of theorem 3. For, if m + n- p > l, w -::;i: 0, !arg 1/wl < (m + n - p- -!)n, lR(,l.) >-! and the parameters dv ... , d 1, av ... , an and bv ... , bm satisfy the conditions (78) and (3), then it follows from assertion 3 of theorem 3 that

(_!_

__1__ Gn,m+! ,1-dv ... , l-d1, l-b 1, .•. , 1-bP) ! p+l,p+l AW ITF(di) 1--av ... , l-aP, l-ev ... , l-c 1 (93)

i-1

--

00

(-1)' r.

~ - - , - !+1'1'! A.

r-o

The function (22) equal to

The function equal to

(-r,d 1, ... ,d1;) an.m+l (110,1-bv ... ,l-bP) . p+l.p+l Cv ... ,c1 ,IjA. w l-av ... ,l-aP,r •

G~·+1~;+ 1 (ljA.w)

G~·+"l_;"£+ 1 (1/w)

on the left of this relation is because of

on the right of (93) is because of (22) and (38)

so that (93) is equivalent to (79) with q = p. 51 )

52) 53 )

Formula (91) follows from (15) and (67). Formula (92) follows from (73). The truth of the Remark may easily be verified.